Mathematical
Theory of Magic Fruits

Interesting
patterns of fractions

In
the year 1560, many warlords were trying to take control
of Japan. At that time, a small region called Owari was
ruled by the Oda clan. Powerful warlords in surrounding
areas were invading Owari, and people thought that the
Oda clan would soon be lost. Nobunaga, who was the young
leader of the Oda clan, secretly summoned the bravest samurais
in his army.

He
told them, "I am going to reveal the utmost secret of our
clan. There is a deep and dangerous cave near the shrine
of our guardian god. Inside the cave there grows a special
fruit tree. Some of its fruits will give you the power
to rule Japan. Other fruits are just bitter. There may
be more than one magic fruit, but once a person eats a
magic fruit and obtains power, the other magic fruits will
turn into bitter fruits".

Problem
1
Only 2 samurais survived to reach the tree. Their name were
Hideyoshi and Mitsuhide. The tree had 6 fruits. The two samurai
wanted the magic fruit very much, but they were wise enough
not to fight for the magic fruit. They took turns and began
to eat the fruits. Since Mitsuhide was older, he ate first.
If there is only one magic fruit, what is the probability
of Mitsuhide's eating it?

Answer
We denote Mitsuhide by 'M' and Hideyoshi by 'H'.
Let's say that M is going to eat first. In the first
round, the probability of M's eating a magic fruit
is 1/6. Let's calculate the probability of M's eating
the magic fruit in the third round. The probability of M's
not eating the magic fruit in the first round is 5/6 and
in the second round H should not eat the fruit.
Its probability is 4/5, since there are only 5 fruits, one
of which is the magic fruit. There are 4 fruits remaining,
including the magic fruit. The probability of M's
eating the magic fruit is 1/4. Therefore, the probability
of M's eating the magic fruit in the third round
is
. As to the probability
of M's eating the magic fruit in the fifth round
we can do almost the same calculation and we get

Therefore
the probability of M's getting the magic fruit
is

We
see then that both M and H have the same
chance of eating the magic fruit.

Problem
2
Suppose that we have the same situation as in Problem 1 except
that there are two magic fruits among the 6 fruits.

Answer
Clearly, the probability of M's eating the magic
fruit in the first round is 2/6, and the probability of his
not eating it is 4/6. We can use almost the same method we
used in Problem 1, and we come out with an answer of

So
far we have studied the case of 6 fruits, but we can change
the numbers. For example, we can study the case of 100
fruits, where 53 of them are magic fruits. We denote by
F[n, m] the probability of the first
man's eating the magic fruit when there are n-fruits and
m-magic ones among them. For example, from Problem 1 we
have F[6,1] = 1/2, and from Problem 2 F[6,2] = 3/5. Similarly
we have F[n, m] =

By
this formula you can calculate F[n, m]
for any natural numbers n and m.

If
you find this formula too difficult to understand, do not
worry about it. If you can understand Problem 1 and 2,
you can understand the rest of our article. You just have
to understand that there is a way to calculate F[n, m]
for any n and m. Now we are going to
show you a new discovery in mathematics! With F[n, m]
for any n and m we make a triangle (see
Figure 1).

Figure
1

By
calculating F[n, m] we obtain a second
triangle (Figure 2) from the above triangle. Let's compare
these two triangles. F[6,3] is the third entry in the 6th
row of the above triangle. In the same position in the
triangle below we have 13/20. Therefore F[6,3]= 13/20.

Figure
2

Problem
3
Can you find any pattern in Figure 2 above?

Answer
Let's compare Figure 2 to Figure 3 below. The pattern is
quite obvious in Figure 3. For example look at 6th row.
F[6,2] = 9/15 and F[6,3] = 13/20 the second and third
entries in the row. F[7,3] = 22/35 =
, which is the third entry in the 7th row, reminds
us of Pascal's
triangle. In general, there exists the same kind of relationship
among F[n, m], F[n, m+1],
F[n+1, m+1] for any natural number n and m with m < n.
We omit the proof.

Figure
3

Problem
4
Can you find any other pattern in Figure 2?

Hint
In fact there are several patterns. Please look at the following
Figure 4, 5 and 6 below. The patterns are obvious, aren't
they?

Figure
4

Figure
5

Figure
6

Remark
We can also study the case with more than two people. For
example, if 4 people play the game, then the probability
of the third person's eating the magic fruit forms the following
triangle. Can you find any pattern in this triangle?

Figure
7

Thanks
We would like to thank Ms. Carolyn Ashizawa for her help
in correcting our English.